具有7-（8,0）型自同构的二元双偶极值码

J4 ›› 2015, Vol. 37 ›› Issue (11): 2142-2147.

具有7-（8,0）型自同构的二元双偶极值码

王荣,王俊新

（山西财经大学应用数学学院，山西太原 030006）

收稿日期:2014-08-11 修回日期:2014-10-28 出版日期:2015-11-25 发布日期:2015-11-25
基金资助:
国家自然科学基金资助项目（70973072，70573066）;山西财经大学青年科研基金资助项目( 晋财大校［2014］90号)

Extremal self-dual doublyeven binary
codes with type of 7-(8,0)

WANG Rong，WANG Junxin

（College of Applied Mathematics,Shanxi University of Finance and Economics，Taiyuan 030006,China）

Received:2014-08-11 Revised:2014-10-28 Online:2015-11-25 Published:2015-11-25

摘要/Abstract

摘要：

把长度为56的具有7（8,0）型自同构的二元双偶极值码看成二元域上的多项式，分解成一个收缩码和一个偶重量多项式的直和。利用已知的码长较短的编码构造出长度为56的具有7（8，0）型自同构的二元双偶极值码，并给出可能的生成矩阵。由于该双偶极值码的最小距离为12，根据两个码等价的类型，对生成矩阵运行Matlab程序，证明了在等价情况下，当E1(σ)*的维数K=2时，不存在有7（8，0）型自同构的二元双偶极值码；当E1(σ)*的维数K=4时，有7（8，0）型自同构二元双偶极值码共75种。对长度为56到70的具有类似型自同构的双偶极值码进行相似讨论，得到了其生成矩阵和分类情况。至此，长度在50到70之间的二元双偶极值码的生成矩阵和分类情况得到完全解决。

关键词: 二元双偶极值码, 自同构, 生成矩阵, 等价, 循环矩阵

Abstract:

By regarding the extremal selfdual doublyeven binary codes with type of 7(8,0) as a polynomial over GF(2)n , we can factorize it to a contract code and even weight polynomials, and construct extremal selfdual doublyeven binary codes with type of 7(8,0) by shorter codes. The possible generator matrix of the codes is given. Matlab procedure implementation shows that, because the minimal distance is 12, codes with type of 7(8,0) do not exist when the dimension of E1(σ)* is 2;and there are 75 selfdual doublyeven binary codes with type of 7(8,0) when the dimension of E1(σ)* is 4. We discuss the similarities of the extremal selfdual doublyeven binary codes with length of 56 to 70, and obtain the generator matrix and classification. So the generator matrices and classification of the extremal selfdual doublyeven binary codes with length between 50 and 70 can be solved completely.Key words:

Key words: extremal selfdual doublyeven binary codes;automorphism;generator matrix;equivalence;cyclic matrix

王荣,王俊新. 具有7-（8,0）型自同构的二元双偶极值码[J]. J4, 2015, 37(11): 2142-2147.

WANG Rong，WANG Junxin. Extremal self-dual doublyeven binary
codes with type of 7-(8,0) [J]. J4, 2015, 37(11): 2142-2147.

参考文献

［1］Ning Ping.Exploration for parallel computing of CRC16 checksum on FPGA［J］.Computer Engineering & Science,2014,36(6):10231027.(in Chinese)
［2］Huffman W C. On the classification and enumeration of selfdual codes ［J］. Finite Fields and Their Appplications,2005,11(3):451490.
［3］Pless V. A classification of selforthogonal codes over GF(2)［J］. Discrete Math,1972,3:209246.
［4］Bouyuklieva S,Yankov N,Kim J.Classification of binary selfdual ［48,24,10］ codes with an automorphism of odd prime order［J］.Finite Fields and Their Applications,2012,18(6):11041113.
［5］AguilarMelchor C,Gborita P,Kim JL,et al. Classification of extremal and sextremal binary selfdual codes of length 38［J］.IEEE Transactions on Information Theory,2012,58(4):22532262.
［6］Yorgov V. The extremal codes of length 42 with automorphism of order 7［J］.Discrete Mathematics,1998,190:201213.
［7］Yorgov V. Erratum to “The extremal codes of length 42 with automorphism of order 7”［J］.Discrete Mathematics,2011,311:18601861.
［8］Bouyuklieva S,Yankov N,Russeva R. Classification of the binary selfdual ［42,21,8］ codes having an automorphism of order 3［J］.Finite Fields and Their Applications,2007,13(3):605615.
［9］Goodwin V,Yorgov V. New extremal selfdual doublyeven binary codes of length 88［J］.Finite Fields and Their Applications,2005, 11(1):15.
［10］Conway J H,Sloane N J A. A new upper bound on the minimal distance of selfdual codes［J］.IEEE Transactions on Information Theory,1990,36(6):13191333.
［11］Yorgov V Y. Binary selfdual codes with automorphism of odd order［J］.Problems of Inform Transmission,1983,19:1124.
［12］Han S,Kim JL,Lee H,et al. Construction of quasicyclic selfdual codes［J］.Finite Fields and Their Applications,2012,18(3):613633.
［13］Bouyuklieva S,Bouyukliev I. An algorithm for classification of binary selfdual codes［J］. IEEE Transactions on Information Theory,2012,58(6):39333940.
［14］Wang Rong,Wang Junxin. ［52，26，10］binary selfdual codes with type of（17，3，1）［J］. Computer Engineering and Applications,2012,48（19）：4648.(in Chinese)
附中文参考文献：
［1］宁平. FPGA上实现CRC16纠错编码并行计算的探讨［J］. 计算机工程与科学,2014,36(6):10231027.
［14］王荣，王俊新. 有（17,3,1）型的二元自对偶编码［52,26,10］［J］.计算机工程与应用,2012,48(19):4648.

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